2.4 Test functions and distributions91D95CD7

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A locally integrable function need not be differentiable, but integration by parts shows how its derivatives should act: if $u\in C^1(\Omega )$ and $\chi $ is smooth with compact support in $\Omega $, then $\int (\partial _ ju)\chi \, \mathrm dx=-\int u\, \partial _ j\chi \, \mathrm dx$, and the right side makes sense for every locally integrable $u$. Distributions are the continuous linear functionals on such functions $\chi $.

A test function on $\Omega $ is an element of $C_ c^\infty (\Omega )$. For a nonempty compact set $K\subset \Omega $, let $\mathcal D_ K(\Omega )$ be the space of test functions supported in $K$. A multi-index is an element $\alpha =(\alpha _1,\ldots ,\alpha _ d)\in \mathbb N^ d$; we write $|\alpha |=\alpha _1+\cdots +\alpha _ d$ and $\partial ^\alpha =\partial _{x_1}^{\alpha _1}\cdots \partial _{x_ d}^{\alpha _ d}$. We give $\mathcal D_ K(\Omega )$ the topology defined by the seminorms

and $\mathcal D(\Omega )=C_ c^\infty (\Omega )$ the topology defined by all seminorms on $C_ c^\infty (\Omega )$ whose restriction to every $\mathcal D_ K(\Omega )$ is continuous.

Here a family $\mathcal S$ of seminorms on a complex vector space $E$ defines the topology in which a set $G\subset E$ is open if for every $\chi _0\in G$ there are $q_1,\ldots ,q_ k\in \mathcal S$ and $r>0$ with $\{ \chi :q_ l(\chi -\chi _0)<r\text{ for }1\leq l\leq k\} \subset G$. A net $(\chi _ i)$ in $E$ converges to $\chi $ for this topology whenever $q(\chi _ i-\chi )\to 0$ for every $q\in \mathcal S$, because every neighborhood of $\chi $ contains a set of the above form. A seminorm $q$ on $E$ is continuous if and only if there are $q_1,\ldots ,q_ k\in \mathcal S$ and $C\geq 0$ with $q\leq C\max _ lq_ l$. Indeed, if $q$ is continuous, the open set $\{ q<1\} $ contains a set $\{ \chi :q_ l(\chi )<r\text{ for }l\leq k\} $. For $\chi \in E$ with $M=\max _ lq_ l(\chi )>0$, the vector $r\chi /(2M)$ belongs to this set, so $q(\chi )<2M/r$; if $M=0$, then $t\chi $ belongs to it for every $t>0$, so $q(\chi )=0$. Conversely, such a bound gives $|q(\chi )-q(\chi _0)|\leq q(\chi -\chi _0)\leq C\max _ lq_ l(\chi -\chi _0)$, and $q$ is continuous. The same argument, applied to $q=|T|$, shows that a linear functional $T$ on $E$ is continuous if and only if $|T|\leq C\max _ lq_ l$ for some $q_1,\ldots ,q_ k\in \mathcal S$ and $C\geq 0$.

Since $p_{K,m}$ increases with $m$, a seminorm on $\mathcal D_ K(\Omega )$ is therefore continuous if and only if it is bounded by a constant multiple of a single $p_{K,m}$. In particular, let test functions $\chi _ j$ be supported in a fixed compact set $K$, and suppose that $\partial ^\alpha \chi _ j\to \partial ^\alpha \chi $ uniformly for every $\alpha $. Then $\chi $ is supported in $K$, and every seminorm $q$ defining the topology of $\mathcal D(\Omega )$ satisfies $q(\chi _ j-\chi )\leq Cp_{K,m}(\chi _ j-\chi )\to 0$ for suitable $C$ and $m$; hence $\chi _ j\to \chi $ in $\mathcal D(\Omega )$.

Definition 2.4.1.55085BC2 A distribution on $\Omega $ is a complex linear functional $\chi \mapsto \langle T,\chi \rangle $ on $C_ c^\infty (\Omega )$ such that for each nonempty compact set $K\subset \Omega $ there are $C>0$ and $m\in \mathbb N$ with

\[ |\langle T,\chi \rangle |\leq C p_{K,m}(\chi )\qquad \text{for all }\chi \in \mathcal D_ K(\Omega ). \]

A distribution $T$ is real if $\langle T,\chi \rangle $ is real for every real test function $\chi $, and a real distribution is positive if $\langle T,\chi \rangle \geq 0$ for every nonnegative test function $\chi $. The space of distributions on $\Omega $ is denoted by $\mathcal D'(\Omega )$.

The constants $C$ and $m$ may depend on $K$. The estimate is equivalent to continuity of $T$ on $\mathcal D(\Omega )$. Indeed, the maximum of finitely many seminorms whose restrictions to every $\mathcal D_ K(\Omega )$ are continuous is again such a seminorm, so by the criterion above, $T$ is continuous if and only if $|T|\leq Cq$ for a single seminorm $q$ defining the topology of $\mathcal D(\Omega )$. If $T$ is continuous, the restriction of this $q$ to $\mathcal D_ K(\Omega )$ is continuous, hence bounded by a multiple of some $p_{K,m}$, and the estimate follows. Conversely, under the estimate, the restriction of the seminorm $\chi \mapsto |\langle T,\chi \rangle |$ to every $\mathcal D_ K(\Omega )$ is continuous, so this seminorm is one of those defining the topology of $\mathcal D(\Omega )$, and $T$ is continuous.

A locally integrable function $u$ defines a distribution by $\langle u,\chi \rangle =\int _\Omega u\chi \, \mathrm d x$; on $\mathcal D_ K(\Omega )$ the estimate holds with $m=0$ and $C=1+\int _ K|u|\, \mathrm d x$. Likewise, a Radon measure defines a distribution by integration (theorem 2.5.6). These pairings are bilinear, without complex conjugation; other conventions are compared in subsection 25.1.1. Changing $u$ on a null set does not change its distribution, and conversely, by proposition 2.5.3 below, the distribution determines $u$ up to equality almost everywhere. A distribution $T$ vanishes on an open set $U\subset \Omega $ if $\langle T,\chi \rangle =0$ whenever $\operatorname{Supp}\chi \subset U$, and its support $\operatorname{Supp}T$ is the complement in $\Omega $ of the union of all open sets on which $T$ vanishes.

For a multi-index $\alpha $, the distributional derivative of $T\in \mathcal D'(\Omega )$ is defined by

Since $p_{K,m}(\partial ^\alpha \chi )\leq p_{K,m+|\alpha |}(\chi )$, an estimate of order $m$ for $T$ on $\mathcal D_ K(\Omega )$ gives one of order $m+|\alpha |$ for $\partial ^\alpha T$. By integration by parts, this definition agrees with the classical derivative for smooth functions. If $a\in C^\infty (\Omega )$, the product $aT$ is defined by $\langle aT,\chi \rangle =\langle T,a\chi \rangle $; it is a distribution because Leibniz's formula bounds the derivatives of $a\chi $ on $K$ by those of $\chi $. Unwinding the definitions gives $\partial _ j(aT)=a\partial _ jT+(\partial _ ja)T$.

Example 2.4.2.E2DD6B809EB8 For $x\in \Omega $, the Dirac measure $\delta _ x$ assigns $1$ to Borel sets containing $x$ and $0$ to the others; as a distribution, $\langle \delta _ x,\chi \rangle =\chi (x)$. In one real dimension, with $0\in \Omega $, we have $\langle \delta '_0,\chi \rangle =-\chi '(0)$. This distribution is not given by integration against a real signed Radon measure in the sense of section 2.2. Indeed, let $\theta $ be a test function with $\theta (0)=1$, for instance a multiple of the function $\rho _\varepsilon $ of section 2.5 with $\varepsilon $ small. The test functions $\theta (x)\sin (nx)$ are bounded by $\sup |\theta |$ and supported in $\operatorname{Supp}\theta $, while their pairings with $\delta '_0$ equal $-\theta (0)n\cos 0-\theta '(0)\sin 0=-n$. Integration against a real signed Radon measure $\sigma $ would bound these pairings by $\sup |\theta |\cdot |\sigma |(\operatorname{Supp}\theta )$, since $|\int f\, \mathrm d\sigma |\leq \int |f|\, \mathrm d|\sigma |$.